Global well-posedness and growth of Sobolev norms for nonlinear Dirac equations with Yukawa potential on compact manifolds

Abstract

In this paper we study a nonlinear Dirac equation with intrinsic Yukawa potential on compact manifolds. In dimension 2, and in dimension 2 after projection onto the positive-energy sector, we prove global well-posedness together with exponential upper bounds on the growth of higher-order Sobolev norms. The main ingredient that allows to extend the local solutions, which are constructed by standard contraction argument, is a commutator-based higher-order energy method, combined with Grönwall's lemma, which provides the necessary a priori Sobolev bounds on the solutions.